The Change of Base Formula merely states that:
log_color(green)(b) color(red)(a) = (log color(red)(a))/(log color(green)(b)) = (ln color(red)(a))/(ln color(green)(b))
(By the way, the equivalence to ln a/ln b just means that lnx/logx is a constant: ~~2.303.)
You just end up taking the base 10 or e logarithm of a and divide it by the same kind of log on b. The base changes so that both bases are the same. For example:
Base 10 counting (decimal system)
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
Base 2 counting (binary system)
1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, ...
1010 in base 2 (you can emphasize this as 1010_2) is equal to 10 in base 10 (you can emphasize this as 10_10).
To get two people to communicate well, they must speak the same language. To be able to divide two logarithms at all, you have to get the numbering systems to be the same.
Conveniently, since (log a)/(log b) = (ln a)/(ln b), you don't even need to specify the base of the "new log". All you do is:
log_2 15 = (log 15)/(log 2) = ln 15 / ln 2 = (ln (3*5))/ln 2 = color(blue)((ln 3 + ln 5) / ln 2 ~~ 3.907)
